The Burtoniana is spread out over several archives and in private collections. An important source of information for Burton MSS held in private hands, and for rare variant editions of his works, are the auction catalogues.
Naturally, many collections of Burton material are in the UK. There is a combined listing of the UK holdings at The National Archives catalog, though note that it is not exhaustive. The National Archives itself holds a great deal of original documents related to Burton's consular career.
The Quentin Keynes Collection of manuscripts is now held at the magnificent British Library and contains many important rarities. The India Office Records, which have extensive details about Burton's career in the Army of the East India Company, are also available there, in the Asian & African Studies Reading Room.
The Wiltshire and Swindon History Centre contains an extensive collection of Arundell family papers, with a great deal of under-utilized Burton material.
The Orleans House Gallery has many items of interest, including some of Burton's personal effects, original portraits, photographs and some correspondence.
The National Library of Scotland has the Grant Papers, which include correspondence between John Hanning Speke and Christopher Palmer Rigby, Speke and Grant, and Grant and Rigby. They also have the John Kirk papers, including some Burton correspondence, and other letters related to Burton.
Many resources are also held in the United States. Burton's personal collection of books, once housed at the Royal Anthropological Institute, are now at Huntington Library in San Marino, California (near Los Angeles). The extensive Eckenstein, Metcalfe, Smithers and Casari Collections are also housed at the Huntington, and are especially important for Burton's later career.
At the Smithsonian in Washington DC, the Russell E. Train Africana Collection contains many rare photographs and letters, most of them now online.
Syracuse University has a lesser known collection.
The Zanzibar Archives apparently have some surviving material relating to the East Africa Expedition of 1856-9.